*The basis for discussion is the abstract of the paper below. Instead of their ‘high-integer near commensurabilities among lunar months’ we’ll just say ‘numbers’ and try to make everything as straightforward as possible. This will expand on a previous Talkshop post on much the same topic.*

Hunting for Periodic Orbits Close to that of the Moon in the Restricted Circular Three-Body Problem (1995)

Authors: G. B. Valsecchi, E. PerozziA, E. Roy, A. Steves

**Abstract**

The role of high-integer near commensurabilities among lunar months — like the long known Saros cycle — in the dynamics of the Moon has been examined in previous papers (Perozzi et al., 1991; Roy et al., 1991; Steves et al., 1993). A by-product of this study has been the discovery that the lunar orbit is very close to a set of 8 long-period periodic orbits of the restricted circular 3-dimensional Sun-Earth-Moon problem in which also the secular motion of the argument of perigee ω is involved (Valsecchi et al., 1993a). **In each of these periodic orbits 223 synodic months are equal to 239 anomalistic and 242 nodical ones, a relationship that approximately holds in the case of the observed Saros cycle**, and the various orbits differ from each other for the initial phases. Note that these integer ratios imply that, in one cycle of the periodic orbit, the argument of perigee ω makes exactly 3 revolutions, i.e. the difference between the 242 nodical and the 239 anomalistic months (these two months differ from each other just for the prograde rotation of ω).

*[bold added]*

To start with we can create a model that pretends the ‘high-integer near commensurabilities’ really are whole numbers, then break down the logic of the result to see what’s going in with the Moon at the period of one Saros cycle.

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